Solving simultaneous equations by elimination

Simultaneous equations are when we have two or more equations with multiple unknowns (letters that represent numbers we don’t know yet). The goal is to find the values of these letters (variables) that make all the equations true at the same time.

When can we solve using elimination?

We can use the elimination method when the equations are both linear (in the form ax + by = c), and we have the same number of equations as unknowns ( for example, two equations with two unknowns, x and y).

Basic steps to solve

The key steps to solve a simultaneous equation with two unknowns by elimination are:

• Find both equations in the form ax + by = c.
• Multiply one or both equations by a number so that the coefficients (the numbers in front of the variables) of one of the variables are the same (or opposites). This allows us to cancel out that variable.
• Add or subtract one equation from another. This involves adding or subtracting each term individually to form a new equation with just one variable.
• Solve this new equation to find the value of one variable.
• Substitute this value back into one of the original equations to find the value of the other variable.

Like with most concepts, it’s much easier to see with some examples.

Examples

Example: solve the simultaneous equations 2x + 3y = 16 and 4x - y = 2.

• First, we have the two equations:
  • 2x + 3y = 16 (Equation 1)
  • 4x - y = 2 (Equation 2)
• We want to eliminate one of the variables. You can do whichever you want, but I will eliminate y here. To do this, we can multiply Equation 2 by 3 so that the coefficient of y in both equations will be opposites:
  • 2x + 3y = 16 (Equation 1)
  • 12x - 3y = 6 (Equation 2 multiplied by 3)
• Add the two equations together to eliminate the y variable:
  • (2x + 12x) + (3y - 3y) = 16 + 6
  • 14x + 0 = 22
  • 14x = 22
• Solve for x:
  • x = \frac{22}{14} = \frac{11}{7}
• substitute x = \frac{11}{7} back into one of the original equations to find y. You can use whichever equation you like: I’ll use Equation 1:
  • 2(\frac{11}{7}) + 3y = 16
  • \frac{22}{7} + 3y = 16
  • 3y = 16 - \frac{22}{7}
  • 3y = \frac{112}{7} - \frac{22}{7}
  • 3y = \frac{90}{7}
  • y = \frac{90}{21} = \frac{30}{7}
• So the solution to the simultaneous equations is:
  • x = \frac{11}{7}
  • y = \frac{30}{7}

Checking your solution

To check if your solution is correct, substitute the values of x and y back into one or both of the original equations to see if they work!